Let A_1 and A_2 be two arithmetic means and G | vedclass

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Question

$a , A _1, A _2, b$ are in A.P.

$d =\frac{b-a}{3} ; A_1=a+\frac{b-a}{3}=\frac{2 a+b}{3}$

$A_2=\frac{a+2 b}{3}$

$A_1+A_2=a+b$

$a, G_1, G_2,

Vedclass · Accepted Answer

$\left(A_1+A_2\right)^2 G_1 G_3$

Vedclass · Answer

$a , A _1, A _2, b$ are in A.P.

$d =\frac{b-a}{3} ; A_1=a+\frac{b-a}{3}=\frac{2 a+b}{3}$

$A_2=\frac{a+2 b}{3}$

$A_1+A_2=a+b$

$a, G_1, G_2, G_3, b 	ext { are in G.P. }$

$r=\left(\frac{b}{a}ight)^{\frac{1}{4}}$

$G_1=\left(a^3 bight)^{\frac{1}{4}}$

$G_2=\left(a^2 b^2ight)^{\frac{1}{4}}$

$G_3=\left(a^3ight)^{\frac{1}{4}}$

$G_1^4+G_2^4+G_3^4+G_1^2 G_3^2=$

$a^3 b+a^2 b^2+a b^3+\left(a^3 bight)^{\frac{1}{2}} \cdot\left(a b^3ight)^{\frac{1}{2}}$

$=a^3 b+a^2 b^2+a b^3+a^2 \cdot b^2$

$=a b\left(a^2+2 a b+b^2ight)$

$=a b(a+b)^2$

$=G_1 \cdot G_3 \cdot\left(A_1+A_2ight)^2$

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2$, $G _3$ be three geometric means of two distinct positive numbers. The $G _1^4+ G _2^4+ G _3^4+ G _1^2 G _3^2$ is equal to

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